Hull operators and interval operators in (L,M)-fuzzy convex spaces

Abstract

Considering L being a continuous lattice and M being a completely distributive De Morgan algebra, several basic notions with respect to (L,M)-fuzzy convex structures in the sense of Shi and Xiu are introduced and their relationship with (L,M)-fuzzy convex structures are studied. Firstly, an equivalent form of (L,M)-fuzzy convex structures in the sense of Shi and Xiu is provided. Secondly, two types of fuzzy hull operators are introduced, which are called (L,M)-fuzzy hull operators and (L,M)-fuzzy restricted hull operators, respectively. It is shown that they can be used to characterize (L,M)-fuzzy convex structures. Finally, fuzzy counterparts of interval operators in the (L,M)-fuzzy case are proposed, which are called (L,M)-fuzzy interval operators. It is proved that there is a Galois correspondence between the category of (L,M)-fuzzy interval spaces and that of (L,M)-fuzzy convex spaces and further the category of arity 2 (L,M)-fuzzy convex spaces can be embedded in the category of (L,M)-fuzzy interval spaces as a fully reflective subcategory.